4π holonomy exists only at the E8 rank · The gap residue is frame-construction dependent
The K-sweep stress test delivers the clearest result to date. Of eight k-NN counts tested (K = 6, 8, 10, 12, 14, 16, 20, 24), only K = 8 produces 4π holonomy (ΔΘ/2π ≈ −2). All other K values give windings between 0 and ±3, with no systematic trend. This confirms the k-NN frame construction is the primary determinant of the holonomy value — the 4π result is K=8 specific, not K-invariant. Whether this specificity reflects E8 geometry (rank = 8 simple roots) or a numerical coincidence is now the central question of the SLH.
Note XXV established a "Triple Lock" at w = 0.98: 4π holonomy, the φ/√2 bandgap ratio, and 0.20% α agreement all coincide. But the 0.0135 residual gap between ΔΘ/2π = −1.9865 and the −2.000 target remains unexplained. Gemini's stress test identified three candidates: (A) k-NN curvature truncation, (B) finite lattice size, (C) a topological invariant of the E8 projection. This note runs all three tests simultaneously.
The raw holonomy at n=31, L=8, w=0.98, K=8, r=5.30 to 15 significant digits:
ΔΘ/2π = −1.986474157898219 ΔΘ = −12.481385241998030 rad gap = 0.013525842101781 gap/π = 0.004305409259958 gap×N = 0.432826947257 (N_ring = 32)
The gap does not simplify cleanly: gap × N = 0.4328 (not an integer), gap × 2N = 0.866 (not an integer). Pattern matching against known constants:
| Candidate | Value | gap / candidate | % off int | note |
|---|---|---|---|---|
| 1/N = 1/32 | 0.031250000 | 0.432827 | 43.3% | |
| 1/(2N) = 1/64 | 0.015625000 | 0.865654 | 13.4% | |
| π/232 | 0.013541348 | 0.998855 | 0.11% | 232 = E8 non-simple roots |
| π/240 | 0.013089969 | 1.033298 | 3.3% | 240 = all E8 roots |
| 1/74 | 0.013513514 | 1.000912 | 0.09% | closest match |
| 1/75 | 0.013333333 | 1.014438 | 1.4% | |
| 1/φ9 | 0.013155618 | 1.028142 | 2.8% | |
| 1/φ8 | 0.021286236 | 0.635427 | 36.5% | |
| 2αphys | 0.014594705 | 0.926764 | 7.3% |
The two nearest candidates — 1/74 (0.09% off) and π/232 (0.11% off) — are tantalisingly close but neither constitutes a clean hit. The 232 figure is physically interesting: it equals E8_roots (240) minus the 8 simple roots, which is exactly the count of E8 neighbours that K=8 frames do not sample. However, the 0.11% gap remains larger than the numerical precision of the computation.
The most important result in this note. Testing K = 6 through K = 24:
| K | Nring | ΔΘ/2π | gap | winding | verdict |
|---|---|---|---|---|---|
| 6 | 32 | −0.705741 | 1.294259 | 2π | wrong |
| 8 | 32 | −1.986474 | 0.013526 | 4π ✅ | ONLY PASS |
| 10 | 32 | +0.944452 | 2.944452 | wrong | fail |
| 12 | 32 | −1.095221 | 0.904779 | 2π | fail |
| 14 | 32 | +0.188590 | 2.188590 | wrong | fail |
| 16 | 32 | +2.555729 | 4.555729 | wrong | fail |
| 20 | 32 | +0.107646 | 2.107646 | wrong | fail |
| 24 | 32 | +0.884717 | 2.884717 | wrong | fail |
The 4π holonomy exists only at K=8. This is the clearest stress test the SLH has yet passed through, and the result is mixed: the signal is not robust across different frame constructions.
Two interpretations are on the table. The cautious reading: the 4π result is a tuning artefact of the specific k-NN count we chose. The E8-motivated reading: K=8 is the correct frame construction because E8 has rank 8 (8 simple roots), and frames built on 8 neighbours naturally capture the 8-dimensional symmetry that the other K values break.
Both interpretations are scientifically legitimate. Resolving them requires deriving K=8 from first principles — not from its output, but from a structural argument about why simple-root frames are the physical choice.
Testing lattice depths L = 6 through L = 11 at fixed K=8, w=0.98:
| L | Nsites | Nring | ΔΘ/2π | gap | note |
|---|---|---|---|---|---|
| 6 | 289 | 32 | −1.986474 | 0.013526 | same as L=8 |
| 7 | 401 | 32 | −2.001188 | 0.001188 | closest to −2.000 ← L=7 crossing |
| 8 | 521 | 32 | −1.986474 | 0.013526 | canonical |
| 9 | 649 | 32 | −1.945870 | 0.054130 | worse than L=8 |
| 10 | 821 | 32 | −1.986474 | 0.013526 | same as L=8 |
| 11 | 965 | 32 | −1.945870 | 0.054130 | same as L=9 |
The L-sweep reveals a clear alternating pattern. Even depths (L = 6, 8, 10) all give identical gap = 0.013526. Odd depths from L=9+ give gap = 0.054130. L=7 is exceptional: gap = 0.001188, the closest approach to the 4π target. The gap does not converge to zero as L grows — this is not a finite-size artefact that will vanish with larger lattice. The parity oscillation (even/odd L) suggests the holonomy is sensitive to the parity of the projection depth.
Note XXI already recorded the L=7 crossing (ΔΘ/2π = −2.001, overshoot 0.06%). The L-sweep confirms this: L=7 is within 0.0012 of the 4π target, closer than any other lattice depth. Yet L=7 has only 401 sites vs L=8's 521. The minimum-gap lattice is smaller than the canonical choice. This strongly suggests the gap is governed by the projection geometry, not by the density of sites.
Sweeping w from 0.960 to 1.000 in steps of 0.002, at fixed K=8, L=8:
| w | Nring | ΔΘ/2π | gap |
|---|---|---|---|
| 0.960 | 28 | +0.000035 | 2.0000 |
| 0.962–0.978 | 28–32 | various | 0.49–2.94 |
| 0.980 | 32 | −1.986474 | 0.01353 |
| 0.982–1.000 | 32–36 | various | 0.75–2.89 |
Only w=0.980 achieves near-4π winding in the entire range 0.960..1.000. The resonance width is less than 0.002 (the step size); the nearest windows (0.978 and 0.982) give gaps of 0.492 and 0.778. This confirms that the w=0.98 resonance is an isolated spike, not a broad basin. Any physical theory must explain not just why w=0.98 resonates, but why the resonance width is so narrow.
The K-sweep result forces a clear statement: the entire chain of results (4π holonomy, Sc = 1.410, α lock at 0.20%) is built on a K=8 frame construction. If we change K, the chain breaks.
This is not automatically a refutation. E8 has rank 8 — its simple root basis is 8-dimensional. A local frame built from the 8 nearest neighbours in the physical 2D projection is arguably the minimal frame consistent with the 8D origin of the lattice. When K=10, the frame includes sites that are "outside" the simple root neighbourhood, mixing in higher-shell E8 geometry. On this reading, K=8 is the canonical frame because it is the E8-minimal frame in the physical plane.
The SLH needs a structural argument for why K=8 frames are the correct choice — independent of the fact that K=8 produces 4π holonomy. One route: show that the k-NN graph at K=8 in the 2D physical plane is the projection of the E8 Voronoi neighbour graph, which has coordination number 8 at each lattice site. If the coordination number of the 2D projection matches the E8 rank, then K=8 is derivable — not assumed.
After this stress test, the inventory is:
| Claim | Status | Comment |
|---|---|---|
| 4π holonomy exists at n=31, w=0.98 | Conditional | Requires K=8; not K-invariant |
| Sc = 1.410 (CV=0.2%) | Conditional | Computed at K=8; must verify at other K |
| αH1 = 0.00731 (0.20% error) | Conditional | Follows from Sc; K=8 dependent |
| w=0.98 is unique resonance | Robust | Sharp spike confirmed <0.002 width |
| Gap = topological invariant | Refuted | Gap is K-dependent; not universal |
| L=7 gives closest approach to −2.000 | Robust | Confirmed across all three experiments |
Two paths diverge from here. The structural path: derive K=8 from the E8 coordination geometry (show K=8 is the Voronoi coordination number of the 2D projection). The empirical path: run Sc and the α-lock probe at K=10, 12, 16 — if Sc remains ~1.410 even when the holonomy breaks, Sc may be the more fundamental quantity.
n=31, w=0.98, r=5.30, αh=0.18, λ=0.145
K-sweep: K ∈ {6, 8, 10, 12, 14, 16, 20, 24}
L-sweep: L ∈ {6, 7, 8, 9, 10, 11}
w-sweep: w ∈ [0.960, 1.000], step 0.002
All holonomy values at full float64 precision (15 significant digits)